#1076542

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,16,12)(18,19)(20,24)(26,30)(28,31);(10,14)(11,16)(17,18,21,19)(20,22,24,23)(25,26)(27,31)(28,32)(29,30);(10,14)(12,15)(20,24)(22,23)(26,30)(28,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19,21,18)(20,23,24,22)(25,27,29,32)(26,31,30,28);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,14,11)(17,20,21,24)(18,22,19,23)(25,28,29,31)(26,32,30,27)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 983738);